An architect is trying to build a cover for two cylinders. He knows the radius of Cylinder 1 is 15 inches and the radius of Cylinder 2 is 16 inches. What is the area of both circles?
3.5*5.5=19.25
The formula to find the area is the length times width
Uhhh uhhh uh. uh. h u h uh u u u u u u u u uh.
Solve cos(4x)-cos(2x)=0 ∀ 0<=x<=2pi ..............(0)
Normal solution:
1. use the double angle formula to decompose, and recall cos^2(x)+sin^2(x)=1
cos(4x)=cos^2(2x)-sin^2(2x)=2cos^2(2x)-1 .................(1)
2. substitute (1) in (0)
2cos^2(2x)-1-cos(2x)=0
3. substitute u=cos(2x)
2u^2-u-1=0
4. Solve for x
factor
(u-1)(u+1/2)=0
=> u=1 or u=-1/2
However, since cos(x) is an even function, so solutions to
{cos(2x)=1, cos(-2x)=1, cos(2x)=-1/2 and cos(-2x)} ...........(2)
are all solutions.
5. The cosine function is symmetrical about pi, therefore
cos(-2x)=cos(2*pi-2x),
solution (2) above becomes
{cos(2x)=1, cos(2pi-2x)=1, cos(2x)=-1/2, cos(2pi-2x)=-1/2}
6. Solve each case
cos(2x)=1 => x=0
cos(2pi-2x)=1 => cos(2pi-0)=1 => x=pi
cos(2x)=-1/2 => 2x=2pi/3 or 2x=4pi/3 => x=pi/3 or 2pi/3
cos(2pi-2x)=-1/2 => 2pi-2x=2pi/3 or 2pi-2x=4pi/3 => x=2pi/3 or x=4pi/3
Summing up,
x={0,pi/3, 2pi/3, pi, 4pi/3}
Answer:
A. x = 1/2
General Formulas and Concepts:
<u>Pre-Algebra</u>
<u>Algebra I</u>
- Terms/Coefficients
- Factoring
- Standard Form: ax² + bx + c = 0
- Solving quadratics
Step-by-step explanation:
<u>Step 1: Define</u>
<em>Identify</em>
4x² + 3 = 4x + 2
<u>Step 2: Solve for </u><em><u>x</u></em>
- [Equality Property] Rewrite in standard form: 4x² - 4x + 1 = 0
- Factor: (2x - 1)² = 0
- Solve: x = 1/2
